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Theorem isfin2 8561
Description: Definition of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin2  |-  ( A  e.  V  ->  ( A  e. FinII 
<-> 
A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y ) ) )
Distinct variable group:    y, A
Allowed substitution hint:    V( y)

Proof of Theorem isfin2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pweq 3958 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
21pweqd 3960 . . 3  |-  ( x  =  A  ->  ~P ~P x  =  ~P ~P A )
32raleqdv 3016 . 2  |-  ( x  =  A  ->  ( A. y  e.  ~P  ~P x ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y )  <->  A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y )  ->  U. y  e.  y ) ) )
4 df-fin2 8553 . 2  |- FinII  =  {
x  |  A. y  e.  ~P  ~P x ( ( y  =/=  (/)  /\ [ C.]  Or  y )  ->  U. y  e.  y ) }
53, 4elab2g 3202 1  |-  ( A  e.  V  ->  ( A  e. FinII 
<-> 
A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2642   A.wral 2793   (/)c0 3732   ~Pcpw 3955   U.cuni 4186    Or wor 4735   [ C.] crpss 6456  FinIIcfin2 8546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ral 2798  df-v 3067  df-in 3430  df-ss 3437  df-pw 3957  df-fin2 8553
This theorem is referenced by:  fin2i  8562  isfin2-2  8586  ssfin2  8587  enfin2i  8588  fin12  8680  fin1a2s  8681
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